Pliska Studia Mathematica Bulgarica

Volume 19, 2009

Proceedings of the XIII International Summer Conference on Probability and Statistics and Seminar on Statistical Data Analysis, SDA, 2008

GUEST EDITOR: L. Mutafchiev

Sofia, 2009

C O N T E N T S

• Mitov, K., Yanev, N. Branching Stochastic Processes: Regulation, Regeneration, Estimation, Applications. (pp. 5-58)
• Atanasov, D. Estimation of IRT Parameters over a Small Sample. Bootstrapping of the Item Responses. (pp. 59-68)
• Atanasov, D., Stoimenova, V., Yanev, N. Offspring Mean Estimators in Branching Processes with Immigration. (pp. 69-82)
• Canto e Castro, L., Dias, S., Da Graga Temido, M. Tail Inference for a Law in a Max-Semistable Domain of Attraction. (pp. 83-96)
• Canto e Castro, L., Da Graga Temido, M. On a Second Order Condition for Max-Semistable Laws. (pp. 97-110)
• Gonzalez, M., Gutierrez, C., Martinez, R., Mota, M. On Y-Linked Genes and Bisexual Branching Processes. (pp. 111-120)
• Dragieva, V. Queue Length Simulations in a Finite Single-line Queueing System with Repeated Calls. (pp. 121-134)
• Jacob, C. Strong Consistency of the Conditional Least Squares Estimator for a Nonstationary Process. Example of the Garch Model. (pp. 135-156)
• Jacob, C., Khraibani, Z., Pancheva, E. Early Detection of Emergent Events Based on an Extremal Process Approach. (pp. 157-172)
• Jordanova, P. Functional Transfer Theorems for Maxima of Stationary Processes. (pp. 173-192)
• Koleva, E., Vuchkov, I., Velev, K. Multiresponse Robust Engineering: Case with Errors in Factor Levels. (pp. 193-206)
• Lio, Y. A Note on Bayesian Estimation for the Negative-Binomial Model. (pp. 207-216)
• Mayster, P. Branching Processes in Autoregressive Random Environment. (pp. 217-228)
• Rahimov, I., Omar, M. Bootstrap for Critical Branching Process with Non-stationary Immigration. (pp. 229-244)
• Shishkov, B., Hashimoto, K., Matsumoto, H., Shinohara, N. Mitani, T. Direction Finding Estimators of Cyclostationary Signals in Array Processing for Microwave Power Transmission. (pp. 245-268)
• Tsitovich, F. Suboptimal Multistage Nonparametric Hypotheses Test. (pp. 269-282)
• Tsitovich, I. Suboptimal Nonparametric Hypotheses Discriminating from Small Dependent Observations. (pp. 283-292)
• Zeifman, A., Chegodaev, A., Satin, Ya. Some Bounds for Almost Absorbing Birth and Death Processes with Catastrophes. (pp. 293-306)
• Ceranka, B., Graczyk, M. Optimum Chemical Balance Weighing Design Under Certain Condition. (pp. 307-318)

A B S T R A C T S

Branching Stochastic Processes: Regulation, Regeneration, Estimation, Applications
Kosto V. Mitov
Nikolay M. Yanev

2000 Mathematics Subject Classification: 60J80
Key words: : Branching process, Regulation, Regeneration, Statistical inference, Applications in biology and economics.

This is a survey of the works of Bulgarian mathematicians in the area of Branching Stochastic Processes.

Estimation Of IRT Parameters Over a Small Sample. Bootstrapping of the Item Responses
Dimitar Atanasov

2000 Mathematics Subject Classification: 97C40
Key words: item response theory, small sample, bootstraping.

Estimation of the parameters of of the Item Response Theory model is reasonable only on a relatively large samples. Applying this methodology for small samples is a common problem in practice. In this paper a bootstrapping technique for a small samples is presented. Additional item responses are added to the original dataset, according to the posterior probability of the correct item response. The same is used in generation of additional items needed when the cognitive attributes are studied.

Offspring Mean Estimators in Branching Processes With Immigration
Dimitar Atanasov
Vessela Stoimenova
Nikolay Yanev

2000 Mathematics Subject Classification: 60J80
Key words: branching processes with immigration, offspring mean estimatimators, simulation.

In the present paper we consider the discrete time branching process with immigration and its relationship to the Bienayme-Galton-Watson process with a random number of ancestors. Several estimators of the offspring mean are considered - the Harris estimator, the conditional least squares estimator of Heyde-Seneta, the conditional weighted least squares estimator of Wei-Winnicki and the estimator of Dion and Yanev. Their properties are compared using computational results based on simulations of the entire immigration family trees. The asymptotic normality of the estimator of Dion and Yanev is combined with the general idea of the trimmed and weighted maximum likelihood. As a result, robust modifications of the offspring mean estimator is proposed.

Tail Inference for a Law in a Max-Semistable Domain of Attraction
Luisa Canto e Castro
Sandra Dias
Maria da Graca Temido

2000 Mathematics Subject Classification: 62G32, 62G05
Key words: max-semistable domain of attraction, geometrically growing sequence, nonparametric estimation.

The class of max-semistable distributions appeared in the literature of extremes, in a work of Pancheva (1992), as the limit distribution of samples with size growing geometrically with ratio r > 1. In Canto e Castro et al. (2002) it is proved that any max-semistable distribution function has a logperiodic component and can be characterized by the period therein, by a tail index parameter and by a real function y representing a repetitive pattern. Statistical inference in the max-semistable setup can be performed through convenient sequences of generalized Pickands' statistics, depending on a tuning parameter s. More precisely, in order to obtain estimators for the period and for the tail index, we can use the fact that the mentioned sequences converge in probability only when s = r (or any of its integer powers), having an oscillatory behavior otherwise. This work presents a procedure to estimate the function y as well as high quantiles. The suggested methodologies are applied to real data consisting in seismic moments of major earthquakes in the Pacific Region.

On a Second Order Condition for Max-Semistable Laws
Luisa Canto e Castro
Maria da Graca Temido

2000 Mathematics Subject Classification: 62G32, 62G20
Key words: max-semistable law, first order condition, second order condition.

In statistics of extremes the great importance of the Normal approximation of intermediate order statistics is well known when the parent distribution function is in a max-stable domain of attraction and verifies the first and the second order extreme value conditions. The generalization of these condi- tions to max-semistable contexts is the object of this paper, aiming to be a basis of future developments in statistical inference under max-semistability.

On Y-Linked Genes and Bisexual Branching Processes
M. Gonzalez
C. Gutierrez
R. Martinez
M. Mota

2000 Mathematics Subject Classification: 60J80
Key words: Sex-linked inheritance. Bidimensional bisexual stochastic model. Perfect fidelity mating. Extinction conditions.

In this paper we survey the results concerning the extinction problem for a two-allele Y-linked gene in a two-sex monogamic population, with a prefer- ence of females for males carrying one of the two alleles of the gene. First we give the mathematical definition of the Y-linked bisexual branching process to model this situation and study some of its relevant properties. Then, we research the extinction of the population and also the survival of each geno- type depending on the behaviour of the other genotype. Finally, we simulate the evolution of the population and conjecture its long term behaviour, for some critical situations.

Queue Length Simulations in a Finite Single-Line Queueing System with Repeated Calls
Velika Ilieva Dragieva

2000 Mathematics Subject Classification: 60K25
Key words: Queueing Theory, Finite Queueing Systems, Repeated Calls, Simulation Experiment.

Simulated results about the queue length and the server state in a finite single server queueing system with repeated calls are presented. Formulas for the basic probability characteristics of the corresponding distributions are obtained in previous papers of the author. The numerical values com- puted according to these formulas are compared with the simulated results. Empirical mean values of the idle period are obtained as well.

Strong Consistency of the Conditional Least Squares Estimator for a Nonstationary Process. Example of the Garch Model
Christine Jacob

2000 Mathematics Subject Classification: 62M10, 62J02, 62F12, 62M05, 62P05, 62P10, 60G46, 60F15.
Key words: Stochastic nonlinear regression, Heteroscedasticity, Nonstationary, Conditional Least Squares Estimator, Consistency, GARCH model.

We consider the Conditional Least Squares Estimator (CLSE) of a unknown parameter θ₀ ∈ R^p of the conditional expectation of a real stochastic process {Y_n} having finite first two conditional moments E(Y_n|F_n-1)< ∞, E(Y_n² | F _n-1)< ∞ at each time n, where E(Yn|Fn-1) is Lipschitz and may be nonlinear in θ₀ and {F_n} is an increasing sequence of σ-algebra. We generalize to this class of processes the necessary and sufficient condition got for the strong consistency of the CLSE of θ₀ in the particular linear deterministic (or linear stochastic if p = 1) model E(Y_n|F_n-1) = θ^T₀W_n. We illustrate this theoretical result with examples, mainly a nonstationary GARCH (1,1) model.

Early Detection of Emergent Events Based on an Extremal Process Approach
Christine Jacob
Zaher Khraibani
Elisaveta Pancheva

2000 Mathematics Subject Classification: 60G70, 62F03
Key words: Extremal process, records, point process, renewal process, sporadic, emergence.

We explore a real renewal process representing the successive arrival times of some event (ex.: clinical case of an infectious disease). We wish to test that the first observed events are sporadic, and not emergent. We also compare this distribution to the one got under the independency of standard setting. We finally illustrate this approach by testing on the first observa- tions of a simulation of a slowly emergent phenomenon that this phenomenon is a sporadic one, and we show that the statistic based on the extremal process is much more efficient and robust than the statistic based on the record values.

Functional Transfer Theorems for Maxima of Stationary Processes
Pavlina Kalcheva Jordanova

2000 Mathematics Subject Classification: 60G70, 60F12, 60G10
Key words: Subexponential distributions, Processes of maxima, Random time, Weak convergence, Stationary sequences.

In this paper we discuss the problem of finding the limit process of sequences of continuous time random processes, which are constructed as properly affine transformedmaxima of randomnumber identically distributed random variables. The max-increments of these processes are dependent. First we work under the well known conditions D (u_n) and D' (u_n) of Leadbetter, Lindgren and Rootzen, (1983). Further we investigate the case of moving average sequence. The distribution function of the noise components is assumed to have regularly varying tails or is subexponential and belongs to the max-domain of attraction of Gumbel distribution or belongs to the max-domain of attraction of Weibull distribution. We work with random time-components which are a.s. strictly increasing to infinity. In particular their counting process is a mixed Poisson process or a renewal process with regularly varying tails with parameter β ∈ (0, 1). Here is proved that such sequences of random processes converges weakly to a compound extremal process.

Multiresponse Robust Engineering: Case with Errors in Factor Levels
Elena Koleva
Ivan Vuchkov
Kamen Velev

2000 Mathematics Subject Classification: 62J05, 62J10, 62F35, 62H12, 62P30
Key words: engineering design, mean and variance models, heteroscedasticity of observations, multiple correlated responses, errors in factor levels, parameter estimation

The model-based robust approach for improving the quality of the process is successfully applied to different industrial processes. In the case of multiple correlated responses the estimation of the mean and variance models of the quality characteristics in production conditions, taking into account the correlation between the multiple responses, together with the heteroscedasticity of the observations due to errors in the factor levels is considered at multivariate regression fit, robust engineering modeling and the optimization stages. The application of the proposed method gives the possibility to use raw industrial data for mean and variance models estimation and leads to reduction of the predicted variance of the responses in production conditions. The proposed approach is applied for electron beam melting and refining experiments.

A Note on Bayesian Estimation for the Negative-Binomial Model
Y. L. Lio

2000 Mathematics Subject Classification: 62F15.
Key words: Negative Binomial Model, Bayes Estimation, Prior Distribution, Posterior Distribution.

The Negative Binomial model, which is generated by a simple mixture model, has been widely applied in the social, health and economic market prediction. The most commonly used methods were the maximum likelihood estimate (MLE) and the moment method estimate (MME). Bradlow et al. (2002) proposed a Bayesian inference with beta-prime and Pearson Type VI as priors for the negative binomial distribution. It is due to the complicated posterior densities of interest not amenable to closed-form integration. A polynomial type expansion for the gamma function had been used to de- rive approximations for posterior densities by Bradlow et al. (2002). In this note, different parameters of interest are used to re-parameterize the model. Beta and gamma priors are introduced for the parameters and a sampling procedure is proposed to evaluate the Bayes estimates of the parameters. Through the computer simulation, the Bayesian estimates for the parame- ters of interest are studied via mean squared error and variance. Finally, the proposed Bayesian estimate is applied to model two real data sets.

Branching Processes in Autoregressive Random Environment
Penka Mayster

2000 Mathematics Subject Classification: 60J80, 60K05
Key words: controlled branching process, state-dependent emigration, random environment, extinction probability, limit theorem in the supercritical case.

We consider the model of alternating branching processes where two Markov branching processes act alternately at random observation and treatment times. The sequences of cycles (observation, treatment) = (δ_n, τ_n) constitute a random environment for branching mechanisms. We suppose in addition that the lengthes of the cycles σ _n = δ_n + τ_n are generated by the linear additive first order autoregressive schema EAR(1).

Bootstrap for Critical Branching Process with Non-Stationary Immigration
I. Rahimov
M. H. Omar

2000 Mathematics Subject Classification: Primary 60J80, Secondary 62F12, 60G99
Key words: branching process, non-stationary immigration, parametric bootstrap, threshold, martingale theorem, Skorokhod space.

In the critical branching process with a stationary immigration the standard parametric bootstrap for an estimator of the offspring mean is invalid. We consider the process with non-stationary immigration, whose mean and variance α(n) and β(n) are finite for each n ≥ 1 and are regularly varying sequences with nonnegative exponents α and β, respectively. It turns out that if α(n) → ∞ and β(n) = o(nα²(n)) as n → ∞, then the standard parametric bootstrap procedure leads to a valid approximation for the distribution of the conditional least squares estimator. We state a theorem which justifies the validity of the bootstrap. By Monte-Carlo and bootstrap simulations for the process we confirm the theoretical findings. The simulation study highlights the validity and utility of the bootstrap in this model as it mimics the Monte-Carlo pivots even when generation size is small.

Direction Finding Estimators of Cyclostationary Signals in Array Processing for Microwave Power Transmission
B. Shishkov
K. Hashimoto
H. Matsumoto
N. Shinohara
T. Mitani

Key words: microwave power transmission, direction finding, cyclostationarity, cyclic second- order statistics, cyclic higher-order statistics, coherent signals, linear prediction signal subspace fitting.

A solar power satellite is paid attention to as a clean, inexhaustible large- scale base-load power supply. The following technology related to beam control is used: A pilot signal is sent from the power receiving site and after direction of arrival estimation the beam is directed back to the earth by same direction. A novel direction-finding algorithm based on linear predic- tion technique for exploiting cyclostationary statistical information (spatial and temporal) is explored. Many modulated communication signals exhibit a cyclostationarity (or periodic correlation) property, corresponding to the underlying periodicity arising from carrier frequencies or baud rates. The problem was solved by using both cyclic second-order statistics and cyclic higher-order statistics. By evaluating the corresponding cyclic statistics of the received data at certain cycle frequencies, we can extract the cyclic corre- lations of only signals with the same cycle frequency and null out the cyclic correlations of stationary additive noise and all other co-channel interfer- ences with different cycle frequencies. Thus, the signal detection capabil- ity can be significantly improved. The proposed algorithms employ cyclic higher-order statistics of the array output and suppress additive Gaussian noise of unknown spectral content, even when the noise shares common cycle frequencies with the non-Gaussian signals of interest. The proposed method completely exploits temporal information (multiple lag ), and also can correctly estimate direction of arrival of desired signals by suppressing undesired signals. Our approach was generalized over direction of arrival es- timation of cyclostationary coherent signals. In this paper, we propose a new approach for exploiting cyclostationarity that seems to be more advanced in comparison with the other existing direction finding algorithms.

Suboptimal Multistage Nonparametric Hypotheses Test
Fedor Tsitovich

2000 Mathematics Subject Classification: 62L10, 62L15
Key words: multistage, sequential methods, hypothesis testing, asymptotic suboptimality, robustness.

At the paper it is considered a discriminating of nonparametric hypotheses that are neighborhoods of given distributions. The suboptimal test means that distributions from the same neighborhoods are indistinguishable. Mul- tistage hypotheses tests have practical advantages over fully-sequential tests in many situations. The suboptimal test with a guaranteed decision is gen- eralized to the multistage case. Using a loss function that is a linear combi- nation of sampling costs and error probabilities, the suboptimal multistage test of nonparametric hypotheses is constructed.

Suboptimal Nonparametric Hypotheses Discriminating from Small Dependent Observations
Ivan Tsitovich

2000 Mathematics Subject Classification: 62L10
Key words: hypothesis testing, suboptimality, small dependence.

It is considered a discriminating of nonparametric hypotheses generated a small dependence of data. The suboptimal test with a guaranteed decision is proposed and numerical results illustrated the procedure suboptimality properties are presented.

Some Bounds for Almost Absorbing Birth and Death Processes With Catastrophes
Alexander Zeifman
Alexander Chegodaev
Yakov Satin

2000 Mathematics Subject Classification: 60J27
Key words: birth and dead process with catastrophes, the rate of convergence, bounds for the mean.

We consider nonstationary almost absorbing birth and death processes (BDPs) with catastrophes. The bounds of the rate of convergence to the limit regime and the estimates of the limit probabilities are obtained. We also study the bounds for the mean of the process and consider a queueing example.

Optimum Chemical Balance Weighing Design Under Certain Condition
Bronislaw Cerankah
Malgorzata Graczyk

2000 Mathematics Subject Classification: 62K05, 05B05.
Key words: Optimum chemical balance, weighing design, ternary balanced block design.

The problem of estimation of unknown weights of p objects is considered. The experiment is carried out according to the standard Gauss-Markoff model of the chemical balance weighing design. Existence conditions of the optimum design are given. New construction method of the optimum design based on the set of the incidence matrices of the ternary balanced block designs is presented.